1. Field of the Invention
The present invention relates to the field of modal analysis and, more particularly, to a method and apparatus for decomposing a complex multiple degree-of-freedom response of a linear dynamic system into individual component single degree-of-freedom modal responses.
2. Description of the Prior Art
Many physical devices are linear dynamic systems. The vibration, or dynamic response, of linear dynamic systems is often the cause of significant problems in many manufacturing processes and inherently limits the ability of certain machines to perform efficiently. The vibration of linear dynamic systems fundamentally limits the accuracy and resolution of sensing systems, while also causing fatigue failure in structural components and electronic assemblies.
Active monitoring and control of vibration is a solution to the aforementioned problems. While researchers have been investigating active vibration control approaches for many years, there still remains fundamental problems which have prevented the widespread deployment of this technology.
Systems which react to a disturbance input by a single-mode response are termed single degree-of-freedom (SDOF) systems. SDOF systems have just a single resonant peak in their frequency response function (FRF) plots and are much easier to control and monitor than multiple degree-of-freedom (MDOF) systems. Unfortunately SDOF systems occur only rarely and are generally achieved only in contrived laboratory experiments.
The dynamic response of a xe2x80x9creal worldxe2x80x9d linear dynamic system is typically a superposition of the response of a plurality of individual modes of response,xe2x80x94i.e. a MDOF system. Real world systems typically have complex response characteristics with many SDOF modes contributing to their dynamic response and many associated modal, or resonant, peaks in their frequency response function (FRFs) as illustrated in FIG. 1A. It has proven difficult to monitor or control systems with such complex response characteristics using prior art techniques.
The general characteristics of modal decomposition are illustrated in FIGS. 1A-1F. These six figures show frequency response functions (FRFs) plotting the input-output relation of a linear system as a function of frequency. A specific example of a linear dynamic system is a mechanical system such as an aircraft. The input to the system is an applied dynamic force and the output is the vibration amplitude measured at different locations on the aircraft. In FIGS. 1A-1F the vertical axis quantity is magnitude displayed in inches per pound force and is defined as the ratio of vibration response amplitude measured at a specific location on the aircraft to the force amplitude applied at a specific location on the aircraft. The horizontal axis quantity is frequency measured in hertz.
FIG. 1A shows a typical FRF measured on a xe2x80x9creal worldxe2x80x9d system comprising a plurality of superposed SDOF modes of vibration. Each peak in the plot of FIG. 1A is associated with a particular SDOF mode of vibration of the linear dynamic system. For example, peak 2 in FIG. 1A is associated with a first SDOF mode of vibration which is shown decoupled in FIG. 1B. Likewise, peak 4 in FIG. 1A is associated with a second SDOF mode of vibration which is shown decoupled in FIG. 1C.
The response of a linear dynamic system is often measured and used by a control system to modify the behavior of the linear dynamic system. As illustrated in FIGS. 1D and 1E, the first and second SDOF modes of vibration may be controlled such that the magnitudes of their respective peaks 2 and 4 are reduced. After implementation of the modal control, the first and second SDOF modes of vibration may be combined to define the complex MDOF response as illustrated in FIG. 1F.
The response of the linear dynamic system may also be observed by an appropriate monitoring device in order to determine the xe2x80x9cstatexe2x80x9d of the linear dynamic system. The monitoring device may detect damage to the linear dynamic system or other changes in operating characteristics by detecting changes in the frequency, amplitude or damping of the SDOF resonant peaks in the FRF plots. Such observation may be improved and simplified by modal filtering, a generally known technique that decomposes the complex MDOF response of a linear system into signals corresponding to the individual constitutive, SDOF modal responses. However, substantial inaccuracies or impracticalities are associated with previously known modal filtering methods. Additionally, prior art modal filtering methods require an excessive number of sensors in order to perform decomposition, cannot account for phase shifts between different sensor data channels and cannot account for different types of sensors being used in combination.
Modal filtering methods are generally well known in the art and are disclosed in numerous publications including Shelley, S. J., Investigation of Discrete Modal Filters for Structural Dynamics Applications, Department of Mechanical and Industrial Engineering, University of Cincinnati, 1990. An improvement to traditional modal filtering techniques, called adaptive modal filtering, utilizes a reference model for calculating modal filter coefficients. Adaptive modal filtering is discussed in many additional publications including, Shelley, S. J., Allemang, R. J., Slater, G. L., Shultze, J. F., Active Vibration Control Utilizing an Adaptive Modal Filter Based Modal Control Method, 11th International Modal Analysis Conference, Kissimmee, Fla., Feb. 1-4, 1993. Both of these documents are incorporated herein by reference. While a significant improvement over prior modal filtering techniques, adaptive modal filtering methods still suffer from many of the disadvantages described above.
Accordingly, there is a need for an improved method and apparatus for decomposing MDOF dynamic responses into signals corresponding to the constituent SDOF modal responses of the MDOF linear dynamic system.
The present invention synthesizes signals corresponding to easily controlled and monitored single degree-of-freedom (SDOF) modal responses by using spatio-temporal filtering to uncouple complicated multiple degree-of-freedom (MDOF) responses measured on real world linear dynamic systems.
The apparatus of the present invention includes at least one sensor, each at least one sensor mounted at a location on a linear dynamic system for generating at least one response isignal representing actual dynamic response of the linear dynamic system at the location. The apparatus also consists of at least one excitation actuator to apply at least one excitation input to the linear dynamic system to generate the dynamic response, and a means for receiving at least one input signal representing at least one excitation input. A data acquisition and processing system periodically samples the response signal at different instances in time, stores a sequence of digitized samples for each sampling of the at least one response signal and associates each digitized sample of the at least one response signal with one of the instances in time. The data acquisition and processing system also periodically samples the at least one excitation input signal in conjunction with the at least one response signal, stores a sequence of digitized samples of the at least one input signal and associates each digitized sample of the at least one input signal with one of the instances in time.
A central data processing unit includes means for reading the digitized samples of the response and input signals and means for calculating therefrom a first set of spatio-temporal filter coefficients. The apparatus further comprises means for generating a first reference model having dynamic characteristics substantially similar to a first single mode of the linear dynamic system, and means for exciting the first reference model in a manner similar to the linear dynamic system thereby producing a first reference modal coordinate response at each of the instances in time.
The first set of spatio-temporal filter coefficients are based upon a plurality of the digitized response signal samples and associated digitized input signal samples. The first set of spatio-temporal filter coefficients have values associated with any one of the instances in time which, when simultaneously applied to a plurality of the digitized samples of the at least one response signal from selected instances in time, will synthesize a signal that substantially matches the first reference modal coordinate response at the one instance in time.
Once the central data processing unit calculates the spatio-temporal filter coefficients a synthesizer applies these coefficients to the response signals. The application of the spatio-temporal filter coefficients to the response signals of the linear dynamic system will synthesize a signal corresponding to a first decoupled SDOF modal response of the linear dynamic system without requiring any additional measurement of the excitation applied to the system As described in the Background of the Invention of the present application, this decoupled SDOF modal response is one of the plurality of individual modal responses which superpose to define the actual MDOF dynamic response of the linear dynamic system. Thus, when the SDOF reference model is excited in a manner similar to the linear dynamic system, the signal synthesized by the spatio-temporal filter corresponds to both the SDOF modal response of the linear dynamic system and also the response of the SDOF reference model.
The central data processing unit further includes means for generating a second reference model having dynamic characteristics substantially similar to a second single mode of the linear dynamic system, and means for exciting the second reference model in a manner similar to the linear dynamic system, thereby producing a second reference modal coordinate response at each of the instances in time. The central data processing unit further comprises means for calculating from the digitized samples of the response signals, a second set of spatio-temporal filter coefficients.
The second set of spatio-temporal filter coefficients are based upon a plurality of the digitized response signal samples and associated digitized input signal samples. The second set of spatio-temporal filter coefficients have values associated with any one of the instances in time which, when simultaneously applied to a plurality of the digitized samples of the at least one response signal from selected instances in time, will synthesize a signal that substantially matches the second reference modal coordinate response at the one instance in time. The synthesizer applies the spatio-temporal filter coefficients to the response signals, thereby generating the synthesized signal corresponding to a second decoupled SDOF modal response of the linear dynamic system.
The central data processing unit may further include means for calculating unlimited additional sets of spatio-temporal filter coefficients in the manner described above with respect to the first and second sets of spatio-temporal filter coefficients. The synthesizer applies the resulting plurality of sets of spatio-temporal filter coefficients to the response signals, thereby generating the synthesized signals corresponding to a plurality of decoupled SDOF modal responses of the linear dynamic system. In this manner, the complex MDOF response of the linear dynamic system may be decoupled into any subset of, or all of its constitutive SDOF modal responses.
The central data processing unit preferably includes means for updating the plurality of sets of spatio-temporal filter coefficients as the response signals are periodically sampled by the data acquisition system.
As noted above, at least one and preferably a plurality of excitation actuators provide at least one excitation input to the linear dynamic system and means are provided for receiving input signals representing at least one excitation input. In the case of multiple excitation inputs the central data processing unit includes means for calculating a set of input influence coefficients based upon the digitized samples of the response and input signals. In such a case the reference model consists of multiple subcomponent reference models, wherein the number of subcomponent reference models is equal to the number of excitation inputs. Each subcomponent reference model is identical to the other, however each is excited in response to a different excitation input signal, and thereby generates a different subcomponent reference modal coordinate response. The set of input influence coefficients have values which, when summed together after being applied individually to the plurality of separate subcomponent reference modal coordinate responses, correspond to an analytical representation of the SDOF modal response of a subject mode of the linear dynamic system. In other words, the input influence coefficients represent the degree to which each input to the system excites the subject mode of the linear dynamic system. Means are provided for generating a set of control force vectors in response to said set of input influence coefficients.
A modal controller is preferably defined by the central data processing unit for generating a modal control signal in response to the first synthesized signal corresponding to the first decoupled SDOF modal response of the linear dynamic system.. A plurality of modal controllers may be defined by the central data processing unit for generating a plurality of modal control signals in response to a plurality of synthesized signals corresponding to a plurality of decoupled SDOF modal responses. In the case of a plurality of excitation actuators the modal control signal is expanded to a plurality of control input signals by multiplying each modal control signal by the set of control force vectors. An actuator power unit independently controls each actuator in response to the control input signals.
The method of the present invention includes the steps of exciting a linear dynamic system, generating at least one excitation input signal representing the actual excitation input to the linear dynamic system, generating at least one response signal representing actual response at a location on the linear dynamic system, periodically sampling the at least one response signal and the at least one excitation input signal to produce a series of digitized samples thereof, and storing the series of digitized samples. The method further comprises the step of processing the digitized samples of the input and response signals to produce a first set of spatio-temporal filter coefficients, the first set being based upon a plurality of the digitized samples of the at least one response signal. The first set of spatio-temporal filter coefficients have values associated with any one of said instances in time which, when simultaneously applied to a plurality of the digitized samples of the response signals, will generate a signal substantially matching a response from a first SDOF reference model being excited in a manner similar to the linear dynamic system. The spatio-temporal filter coefficients are applied to the digitized samples of response signals, thereby generating the synthesized signal corresponding to the decoupled SDOF modal response of the linear dynamic system.. The set of spatio-temporal filter coefficients are updated as the response signals are periodically sampled. The method may further comprise the steps of processing the digitized samples to simultaneously calculate a plurality of sets of spatio-temporal filter coefficients and then applying the coefficients to the response signals.
The step of exciting the linear dynamic system comprises applying at least one excitation input to the linear dynamic system. In the case of multiple excitation inputs the method of the present invention further comprises the steps of generating a set of input influence coefficients having values which, when summed together after being applied individually to the separate subcomponent reference modal coordinate responses, forms an analytical representation of the SDOF modal response of a subject mode of the linear dynamic system. A set of control force vectors are generated in response to the set of input influence coefficients. A modal control signal is generated in response to the first SDOF system response and expanded into a plurality of control input signals by multiplying the modal control signal by the set of control force vectors. The excitation inputs are controlled independently in response to the control input signals.
Therefore, it is an object of the invention to provide a method and apparatus for decomposing with great accuracy complex MDOF responses of linear dynamic systems into synthesized signals corresponding to decoupled constituent SDOF modal responses of the linear dynamic system.
It is a further object of the invention to provide such a method and apparatus requiring a minimal number of sensors in order to decompose the MDOF response.
It is another object of the invention to provide such a method and apparatus which accommodates different types of sensors and phase shifts in the sensors.
It is still yet another object of the invention to provide a method and apparatus for calculating a plurality of spatio-temporal filter coefficients from a plurality of time delayed successive samples of response signals representing actual responses of the linear dynamic system and successive samples of at least one excitation input signal.
It is another object of the invention to provide a method and apparatus for calculating spatio-temporal filter coefficients which decompose the complex MDOF response of a linear dynamic system while accounting for multiple input forces exciting the linear dynamic system.
It is a further object of the invention to provide a method and apparatus for calculating a plurality of input influence coefficients from a plurality of time delayed successive samples of response signals representing responses of the linear dynamic system and a plurality of samples of excitation signals representing the excitation inputs to the linear dynamic system.
Other objects and advantages of the invention will be apparent from the following description, the accompanying drawings and the appended claims.